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# Timothy J. Healey

Professor

### Departments/Programs

- Mathematics

### Graduate Fields

- Applied Mathematics
- Mathematics
- Theoretical and Applied Mechanics

## Research

Applied analysis and partial differential equations, mathematical continuum mechanics

I work at the interface between the nonlinear analysis of PDE and the mechanics of elastic structures and materials. Although not as well-known or popular as mathematical fluid mechanics, nonlinear finite-deformation elasticity is the central model of continuum solid mechanics. While the mathematical theory of linear(ized) elasticity is a classical, well-honed subject, properly formulated problems of the nonlinear theory lead to open mathematical questions. For example, a systematic approach to the construction of weak solutions is still not known to this day.

The two main goals of my work are to establish rigorous existence results and to uncover new phenomena. Along with my co-workers and students, we develop and employ topological-degree-theoretic methods, direct methods of the calculus of variations, and global symmetry-breaking bifurcation theory to establish existence theorems. Examples illustrating the latter goal include things like wrinkling behavior in thin sheets and icosahedral pattern formation in 2-phase lipid bilayer structures. The work involves a symbiotic interplay between three key ingredients: careful mechanics-based modeling, mathematical analysis, and efficient computation.

## Courses

### Spring 2020

- MATH 4280 : Introduction to Partial Differential Equations
- MATH 4900 : Supervised Research
- MATH 4901 : Supervised Reading
- MATH 6160 : Partial Differential Equations

## Publications

- Classical solutions in the large in incompressible nonlinear elasticity, Arch. Rational Mech. Anal. (2018) DOI 10.1007/s00205-018-01342-y.
- Direct construction of symmetry-breaking directions in bifurcation problems with spherical symmetry (with S. Dharmavaram), Discrete and Cont. Dynamical Systems S, 12 (2019) 1669-1684.
- The Mullin effect in the wrinkling behavior of highly stretched thin films (with E. Fejér and A. Sipos), J. Mech. Phys. Solids 119 (2018) 417-427.
- Injectivity and self-contact in second-gradient nonlinear elasticity (with A. Palmer), Calc. Var. PDE 56 (2017) no. 114, DOI 10.1007/s00526-017-1212-y.
- Symmetry-Breaking Global Bifurcation in a Surface Continuum Phase-Field Model for Lipid Bilayer Vesicles (with S. Dharmavaram), SIAM J. Math. Anal. 49 (2017) 1027–1059.
- Direct computation of two-phase icosahedral equilibria of lipid bilayer vesicles (with Q. Li and S. Zhao), Comput. Methods Appl. Mech. Engrg. 314 (2017) 164–179.
- Stability boundaries for wrinkling in highly stretched elastic sheets (with Q. Li), J. Mech. Phys. Solids 97, (2016) 260-274.
- Injective weak solutions in second-gradient nonlinear elasticity (with S. Krömer), ESAIM: COCV 15 (2009) 863–871.
- Material symmetry and chirality in nonlinearly elastic rods, Math. Mech. Solids 7 (2002) 405–420.
- Global continuation in nonlinear elasticity (with H. Simpson), Arch. Rational Mech. Anal. 143 (1998) 1–28.
- Preservation of nodal structure on global bifurcating solution branches of elliptic equations with symmetry (with H. Kielhöfer), JDE 106 (1993) 70-89